D IGITAL L OGIC Number Systems By: Safwan Mawlood Digital Principles and Logic Design, A.Saha &N.Manna

B OOLEAN A LGEBRA Boolean algebra finds its most practical use in the simplification of logic circuits The switching functions can be expressed with Boolean equations. Complex Boolean equations can be simplified by a new kind of algebra, which is popularly called Switching Algebra or Boolean Algebra, invented by the mathematician George Boole in 1854.

ALGEBRAICSTRUCTURES Closer: A set is closed with respect to a binary operator if, for every pair of elements of S, the binary operator specifies a rule for obtaining a unique element of S. X + Y = Z Y + X = Z

Associative Law: A binary operator * on a set S is said to be associated whenever (A*B)*C = A*(B*C) for all A,B,C ∈ S. Commutative Law: A binary operator * on a set S is said to be commutative whenever A*B = B*A for all A,B ∈ S.

Identity Element: A set S is to have an identity element with respect to a binary operation * on S, if there exists an element E ∈ S with the property E*A = A*X = A. Example: The element 0 is an identity element with respect to the binary operator + on the set of integers I = {.... –4, –3, –2, –1, 0, 1, 2, 3, 4,....} as A + 0 = 0 + A = A. Similarly, the element 1 is the identity element with respect to the binary operator × as A × 1 = 1 × A = A.

Inverse: If a set S has the identity element E with respect to a binary operator *, there exists an element B ∈ S, which is called the inverse, for every A ∈ S, such that A*B = E. Example: In the set of integers I with E = 0, the inverse of an element A is (-A) since A + (–A) = 0.

Distributive Law: If * and (.) are two binary operators on a set S, * is said to be distributive over (.), whenever A*(B.C) = (A*B).(A*C).

OR AND NOT

D E M ORGAN ' S T HEOREM Two theorems that were proposed by DeMorgan play important parts in Boolean algebra. The first theorem states that the complement of a product is equal to the sum of the complements. That is, if the variables are A and B, then (A.B)' = A' + B‘ The second theorem states that the complement of a sum is equal to the product of the complements. In equation form, this can be expressed as (A + B)' = A'. B'

SIMPLIFICATION OF BOOLEAN EXPRESSIONS Example. Simplify the Boolean function F=AB+ BC + B'C. Solution. F = AB + BC + B'C = AB + C(B + B') = AB + C

T RUTH T ABLE F=AB+C

S IMPLIFICATION B OOLEAN A LGEBRA